Inductive dimension
In the small and large inductive dimension is two considered in the mathematical branch of topology dimension terms. These terms do not use any algebraic constructions for defining a dimension, as it is known for example from the theory of vector spaces, but only itself the considered topological space There is an alternative to the Lebesgue covering dimension, with designated and used here for comparison purposes will.
- 4.1 Compare
- 4.2 compactification
- 4.3 subset set
- 4.4 Summary Record
- 4.5 Product Set
Motivation
The idea of inductive dimension is based on the observation that the edge is one -dimensional ball -dimensional, where- dimensional here in the sense of differential geometry (see diversity ) or is just purely descriptive to understand. This suggests the idea of the notion of dimension n due a lot to the concept of dimension n -1 the edge of this amount and so to seek an inductive definition.
Since a einpunktiger space, which ensures to get the dimension 0, an empty edge, you have to specify the dimension of the empty set as -1. An implementation of the idea of the inductive definition then leads to the following two variants:
Definition
The small inductive dimension
The inductive small dimension of a topological space are defined as follows:
- , If for every point and every open neighborhood of an open neighborhood of is with and.
Thus, explains what this means. We define further:
- If not
- If the inequality holds for any.
The large inductive dimension
If one replaces the point from the definition of small inductive dimension by any closed set, we obtain the notion of large inductive dimension. Specifically, the large inductive dimension of a topological space is defined as follows:
- If for any closed set and every open neighborhood of an open neighborhood of is with and.
Thus, explains what this means. We define further:
- If not
- If the inequality holds for any.
Comments
- As in the one-point subsets spaces are complete, it follows for such rooms immediately.
- Is a discrete space, then.
- The statement can be rephrased as follows: Each point has a neighborhood base of closed sets with edges of the small inductive dimension. In particular, in this case, each point has a neighborhood base of closed sets, so that this term is only meaningful in regular rooms.
- While in the small inductive dimension of each point in the space can be assigned to a dimension in an obvious way, this is not possible with the large inductive dimension, which refers to the overall space.
Theorems on inductive dimension
Comparisons
If a metric space, then by a set of M. Katětov
.
A set of PS Aleksandrov states for compact Hausdorff spaces:
.
Equality has for separable metrizable spaces:
.
K. Nagami has constructed a regular room, for the, and is.
Compactification
Denote the Stone - Čech compactification of. Then we have
- N. Wendenisow: Is normal, so true.
- J. R. Isbell: Is normal, so true.
- A similar statement for the small inductive dimension is wrong.
Subset set
And satisfy the subset rate for totally normal spaces, ie
- Is totally normal and so is true (respectively).
Summary record
The large inductive dimension satisfies the sum theorem for completely regular spaces, ie
- CH Dowker: Is completely normal, and a sequence of closed sets with so true.
- For general normal spaces of the sum rate does not apply for yet, not even if one restricts to compact Hausdorff spaces.
Product set
It is said that a concept of dimension satisfies a product set, if the dimension of the product space of two rooms to the sum of the dimensions of these two spaces can be estimated. Notice.
- Are not empty regular Hausdorff spaces, the following applies.
- Are perfectly normal and metrizable and both non- empty, then applies.
- For the covering dimension applies a similar statement when and both are metrizable or are paracompact and compact.
Swell
- Keiô Nagami: Dimension Theory. Academic Press, New York, NY, among others, 1970, ISBN 0-12-513650-1 ( Pure and Applied Mathematics 37).
- Set topology